| L(s) = 1 | + (−0.786 − 0.618i)2-s + (−1.57 + 2.21i)3-s + (0.235 + 0.971i)4-s + (3.69 − 0.712i)5-s + (2.60 − 0.765i)6-s + (−0.836 − 2.50i)7-s + (0.415 − 0.909i)8-s + (−1.43 − 4.14i)9-s + (−3.34 − 1.72i)10-s + (0.540 − 0.425i)11-s + (−2.52 − 1.00i)12-s + (3.73 − 2.39i)13-s + (−0.893 + 2.49i)14-s + (−4.25 + 9.30i)15-s + (−0.888 + 0.458i)16-s + (4.26 + 4.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (−0.909 + 1.27i)3-s + (0.117 + 0.485i)4-s + (1.65 − 0.318i)5-s + (1.06 − 0.312i)6-s + (−0.316 − 0.948i)7-s + (0.146 − 0.321i)8-s + (−0.477 − 1.38i)9-s + (−1.05 − 0.545i)10-s + (0.162 − 0.128i)11-s + (−0.728 − 0.291i)12-s + (1.03 − 0.665i)13-s + (−0.238 + 0.665i)14-s + (−1.09 + 2.40i)15-s + (−0.222 + 0.114i)16-s + (1.03 + 0.986i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.998483 + 0.0569908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.998483 + 0.0569908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (0.836 + 2.50i)T \) |
| 23 | \( 1 + (-2.86 + 3.84i)T \) |
| good | 3 | \( 1 + (1.57 - 2.21i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-3.69 + 0.712i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-0.540 + 0.425i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.73 + 2.39i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.26 - 4.06i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.60 - 2.47i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (6.22 - 1.82i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-6.71 + 0.641i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-1.97 - 5.70i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (0.543 + 0.627i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.45 - 9.74i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-3.10 + 5.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0866 - 1.81i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (11.6 + 6.01i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (3.68 + 5.17i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-0.228 + 0.0914i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.399 - 2.77i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.743 + 3.06i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.137 - 2.87i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-3.97 + 4.58i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.22 + 0.689i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (10.5 + 12.1i)T + (-13.8 + 96.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07590483335620443555157978207, −10.50042505762168641473914973806, −10.03616448543592254858398600567, −9.316100036036663494589273594273, −8.189099658448333026948407896660, −6.34898516365375989267349807392, −5.81193666195417820223574236852, −4.53554711242902650330839785155, −3.35455044344865479842844069258, −1.24364790996273101216844840995,
1.35900485661441301118081746791, 2.47110891965675781406881176149, 5.38851395981137942739751599443, 5.94734038225704546632855419224, 6.58731461673237264424680450925, 7.46993339393211106966946567965, 8.977670775478784157102803373627, 9.512061462122592444214766872461, 10.73453659991815915924464385690, 11.58869682032274329455242010504
